Models survival times of exponential, Weibull, extreme-value, log-logistic or lognormal distributions (R.W. Payne & D.A. Murray).

### Options

`PRINT` = string tokens |
Controls printed output (`model` , `deviance` , `summary` , `estimates` , `correlations` , `fittedvalues` , `accumulated` , `loglikelihood` ); default `mode` , `summ` , `esti` |
---|---|

`TIMES` = variate |
Time of each observation |

`DISTRIBUTION` = string token |
Distribution of the survival times (`exponential` , `weibull` , `extremevalue` , `loglogistic` , `lognormal` ); default `expo` |

`CENSORED` = variate |
Indicator for censored observations: 0 if uncensored, 1 if right censored (subject survived the whole trial), -1 if left censored (log-logistic distribution only); default assumes no censored observations |

`PLOT` = string token |
What to plot (survivorfunction); default `*` |

`GRAPHICS` = string token |
Type of graphics (`lineprinter` , `highresolution` ) default `high` |

`ALPHA` = scalar |
Saves the estimated value of the parameter α of the Weibull and extreme-value distributions, if the scalar is input with a non-missing value this provides the initial estimate for α (which will also be the final estimate if `MAXCYCLE` =1) |

`_2LOGLIKELIHOOD` = scalar |
Saves -2 multiplied by the log-likelihood |

`SIGMA` = scalar |
Saves the estimated value of the shape parameter sigma of the log-logistic and lognormal distributions |

`SURVIVOR` = variate |
Saves estimates of the survivor function |

`PARAMETERIZATION` = string token |
Controls the parameterization used when saving the survivor function for the Weibull distribution (`ph` , `aft` ); default `ph` |

`MAXCYCLE` = scalar |
Maximum number of iterations to use to estimate α; default 20 |

`TOLERANCE` = scalar |
Convergence limit for α; default 10^{-5} |

### Parameter

`TERMS` = formula |
Defines the model to fit |
---|

### Description

`RSURVIVAL`

models survival times assuming that they follow either an exponential, Weibull, extreme-value, log-logistic or lognormal distribution, as indicated by the `DISTRIBUTION`

option. It also caters for right-censored observations, where the subject concerned survived the trial: the `CENSORED`

option can be used to specify a variate with an entry for each subject containing one where the subject survived, otherwise zero. The log-logistic caters for left-censored observations, which they can be specified by an entry of -1 in the `CENSORED`

variate. The model to be fitted to the survival times is specified using the `TERMS`

parameter.

The analysis is performed using the generalized linear models facilities of Genstat. For the exponential, Weibull and extreme-value distributions a y-variate (= 1 – `CENSORED`

) is specified indicating whether the subject died or survived, and an offset variate is included which depends on the time variate (see Chapter 6 of Aitkin *et al*. 1989). For the exponential distribution this offset is simply the logarithm of the times. With the Weibull distribution it is the Weibull parameter α multiplied by the logarithm of the times, while for the extreme-value distribution it is the parameter α multiplied by the times. The parameters of the `TERMS`

model and α itself are estimated alternately (with number of cycles controlled by the `MAXCYCLE`

option) until successive estimates are within a tolerance specified by the `TOLERANCE`

option. The `ALPHA`

option can input an initial value for α and save the estimated value. By setting the `MAXCYCLE`

option to one, α can be fixed at the initial value; this is useful for comparing one model with another, when the value of α should be fixed at the value estimated from the more complicated model. The log-logistic distribution is fitted using a logistic regression model with number of successes 1-*c* and binomial denominator 2-*c*–*b* (where *c* is an index for a right-censored observation and *b* is an index for a left-censored observation) using an offset variate of the logarithm of times divided by σ. The parameters of the `TERMS`

model and σ (shape parameter) are estimated alternately (with number of cycles controlled by the `MAXCYCLE`

option) until successive estimates are within a tolerance specified by the `TOLERANCE`

option. For the lognormal distribution maximization of the log-likelihood is achieved using an EM algorithm details of which are given in Section 6.19 of Aitkin *et al*. (1989). The `SIGMA`

option can be used to save the estimated value of the shape parameter for both the log-logistic and lognormal distributions. The importance of variables in the lognormal model should be assessed by omitting the variable and comparing -2 times the log-likelihood; this can be saved using the `_2LOGLIKELIHOOD`

option. The `SURVIVOR`

option allows you to save estimates of the survivor function. For the Weibull distribution the `PARAMETERIZATION`

option can be used to choose whether to produce the estimates for the survivor function using the proportional hazards or accelerated failure time parameterization.

The `PRINT`

option controls printed output with similar settings to those of the `FIT`

directive, except that there is an extra setting `loglikelihood`

to print -2 times the log-likelihood. Further information can be printed subsequently by using `RDISPLAY`

in the usual way. The `PLOT`

option can be set to `survivorfunction`

to produce plots of the empirical survivor function against the value predicted by the model, when the exponential, Weibull and extreme-value distributions are selected (see Aitken *et al*. 1989, pages 275-276). The `GRAPHICS`

option determines the type of graph, with settings `highresolution`

(the default) or `lineprinter`

.

Options: `PRINT`

, `TIMES`

, `DISTRIBUTION`

, `CENSORED`

, `PLOT`

, `GRAPHICS`

, `ALPHA`

, `_2LOGLIKELIHOOD`

, `SIGMA`

, `SURVIVOR`

, `PARAMETERIZATION`

, `MAXCYCLE`

, `TOLERANCE`

.

Parameter: `TERMS`

.

### Method

Full details of the method can be found in Chapter 6 of Aitkin *et al*. (1989). For the exponential distribution (pages 269-270), the survivor function is

S(t) = exp(-λ *t*)

with

λ = exp( Σ( *b _{i}*

*x*) )

_{i}where *b _{i}* are the parameter estimates,

*x*are the appropriate values of the

_{i}explanatory variates, and *t* is the time. The Weibull distribution (page 280)

is defined with density function

f(*t*) = α λ *t*^{(α-1)} exp(-λ (*t*^{α}))

and has survivor function

S(*t*) = exp(-λ *t*^{α}).

The extreme-value distribution (pages 283-284) has survivor function

S(*t*) = exp(-λ exp(α*t*)).

The loglogistic distribution (pages 295-297) has the survivor function

S(*t*) = 1 / { 1 + (*t* / θ)^{a} }

with

θ = exp(∑(*b _{i}* ×

*x*))

_{i}and *a* = 1 / σ.

The lognormal distribution (pages 297-300) has survivor function

S(*t*) = `CUNORMAL`

( log( (*t* – ∑(*b _{i}* ×

*x*)) / σ))

_{i}### Action with `RESTRICT`

The vectors involved in the analysis may be restricted as usual for a generalized linear model.

### Reference

Aitkin, M., Anderson, A., Francis, B. & Hinde, J. (1989). *Statistical Modelling in GLIM*. Oxford University Press.

### See also

Procedures: `KAPLANMEIER`

, `RLIFETABLE`

, `RPHFIT`

, `RPROPORTIONAL`

, `RSTEST`

.

Commands for: Survival analysis.

### Example

CAPTION 'RSURVIVAL example',\ 'Data from Gehan (1965, Biometrika, 52, 203-223).'; STYLE=meta,plain VARIATE [VALUES=1,1,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,15,17,22,23,\ 6,6,6,6,7,9,10,10,11,13,16,17,19,20,22,23,25,32,32,34,35] Time & [VALUES=24(0),1,0,1,0,1,1,0,0,1,1,1,0,0,1,1,1,1,1] Censor FACTOR [LABELS=!t(control,'6-mercaptopurine'); VALUES=21(1,2)] Treat PRINT 'Exponential distribution' RSURVIVAL [TIMES=Time; CENSORED=Censor] Treat PRINT 'Weibull distribution' RSURVIVAL [DIST=weibull; TIMES=Time; CENSORED=Censor] Treat